The Deming Cycle has an inherent circular paradigm, it assumes that everything starts with Planning. Plan has a limited range of meaning. Shewart intended that experiments and quality control should be planned to deliver results in accordance with the specifications (see meaning above), which is good advice. However, Planning was not intended to cover aspects such as creativity, innovation, invention or Complex Adaptive Systems. In these aspects particularly when based upon imagination, it is often impossible or counterproductive to plan (see referenced Wikipedia pages for why this is so). Hence, PDCA is inapplicable in these situations.
The Deming Cycle approaches often do not get to the root cause of a problem, especially in adaptive situations which call for an experiential approach but demand much more rigour in analysis and data collection. An adaptive challenge exists where there are no visible solutions to problems, and can exist, for example in areas where chaos, uncertainty, and ambiguity exists, such as new frontiers, and existing complex systems such as Healthcare.
Do and Act have the same meaning in English. Dictionaries (Shorter Oxford) provide the following relevant definitions:
The 'Act' in the Deming Cycle is meant to be interpreted to have a different meaning to 'Do', otherwise it could be as easily have been PDCD or PACA. In PDCA, 'Act' is meant to apply actions to the outcome for necessary improvement (see meaning above), in other words 'Act' means 'Improve' (applying PDCA to itself could result in PDCI).